Erdős' Anti-Calculus Proposition

Proposition

Erdös contends that in Complex Analysis, versus Single Variable Calculus, a function will not have a derivative of zero when it reaches a maximum value.

In Single Variable Calculus:

A function $f(x)$, which is smooth (continuous and differentiable) on a open domain $D$, will have a derivative of zero when it attains a maximum on $D$.

However, for a complex function $g(z)$, on a closed domain $U$ with border $B$, the maximum (Guaranteed by the Extreme Value Theorem), may not have a derivative of zero.

Proof

Let there exist a complex $\epsilon$ such that $|\epsilon| > 0$ and $\delta$ such that $|\delta| > 0$ where $\epsilon$ and $\delta$ are arbitrarily small.

Furthermore, let the complex function $f(z)$ be analytic and non-constant on an open domain $D$, bounded by a contour $B$ along which $f(z)$ is continuous and differentiable. Thus, $f$ is continuous and differentiable on the closed region $U = D \cup B$.

By the Maximum Modulus Theorem, the maximum of $f$ occurs on $B$ at $z_0$.

Now, suppose that $f'(z_0)$ is indeed zero. This would mean that for some $\epsilon$, there is are points $w = z_0 + \delta$ where $\delta < \epsilon$ for which $f(w)=f(z_0)$, since $f(z)$ is continuous. $z_0 + \delta$ is a maximum point for $f(z)$ on $D$. By the Maximum Modulus Theorem, this means that $f$ must be constant on $D$.

Since $f(z)$ is non-constant on $D$, this is a contradiction. $#$